How does the scalarproduct look like, popular kernel functions compute

So far as I understood kernels, they compute a dot product of the vectors

$k(x_i,x_j) = \langle\phi(x_i),\phi(x_j)\rangle$

where $\phi$ goes from original lower dimensional space to a higher dimensional space. The most popular ones are the polynomial Kernel, the radial basis function kernel and the sigmoid kernel.

Regarding this, I have three questions:

1. For each of these kernels, what is actually the function $\phi$ look like (in dependence of the hyperparameters)?

2. Assuming there is a nonlinear decision boundary, that seperates the dataset perfectly with given features, does there always exist a kernel, so that we find this decision boundary with this kernel and SVM?

3. What would be an example of an Dataset (which actually has a nonlinear decision boundary, sperating this set perfectly), where one of those 3 popular kernels fails?

• For 2&3, what if $x_i = x_j$ but $y_i \neq y_j$? In this case, it's impossible to find a boundary to separate the two classes with any kernel. – Sycorax Dec 21 '16 at 16:23
• Ok, excluding this special case :D – Luca Thiede Dec 21 '16 at 16:26
• "Are there any black swans?" "How about this one." "I mean except for that one." – Sycorax Dec 21 '16 at 17:15
• Than let me reformulate my question: Assuming there is a nonlinear decision boundary, that seperates the dataset perfectly with given features, does there always exist a kernel, so that we find this decision boundary with this kernel and SVM? – Luca Thiede Dec 22 '16 at 9:35
• Same for 3. just considering datasets, where a perfect (perfect under the given dataset) decision boundary actually exists – Luca Thiede Dec 22 '16 at 9:37